Two universal results for Wilson loops at strong coupling

Recently it has been proposed that Wilson loops in high-dimensional representations in N=4 supersymmetric Yang-Mills theory (or multiply wrapped loops) are described by D-branes in AdS_5 x S^5, rather than by fundamental strings. Thus far explicit D3-brane solutions have been only found in the case of the half-BPS circle or line. Here we present D3-brane solutions describing some 1/4 BPS loops. In one case, where the loop is conjectured to be given by a Gaussian matrix model,...

We prove that the half-BPS Wilson loop operator of N=4 SYM in a symmetric representation of the gauge group has a bulk gravitational description in terms of a single D3-brane in AdS_5xS^5, as argued in hep-th/0604007. We also show that a half-BPS Wilson loop operator in an arbitrary representation is described by the D3-brane configuration proposed in hep-th/0604007. This is demonstrated by explicitly integrating out the degrees of freedom on the D3-branes and showing that th...

Based on an extension of the holographic principle to superspace, we provide a strong-coupling description of smooth super Wilson loops in terms of minimal surfaces of the $AdS_5 \times S^5$ superstring. We employ the classical integrability of the Green-Schwarz superstring on $AdS_5 \times S^5$ to derive the superconformal and Yangian $Y[\mathfrak{psu}(2,2|4)]$ Ward identities for the super Wilson loop, thus extending the strong coupling results obtained for the Maldacena-Wi...

We study the action of S-duality on half-BPS Wilson loop operators in 5d $\mathcal{N}=1$ theories. The duality is the statement that different massive deformations of a single 5d SCFT are described by different gauge theories, or equivalently that the SCFT points in parameter space of two gauge theories coincide. The pairs of dual theories that we study are realized by brane webs in type IIB string theory that are S-dual to each other. We focus on $SU(2)$ SQCD theories with $...

We study the half-BPS circular Wilson loop in ${\cal N}=4$ super Yang-Mills with orthogonal gauge group. By supersymmetric localization, its expectation value can be computed exactly from a matrix integral over the Lie algebra of $SO(N)$. We focus on the large $N$ limit and present some simple quantitative tests of the duality with type IIB string theory in $AdS_5\times \mathbb{RP}^5$. In particular, we show that the strong coupling limit of the expectation value of the Wilso...

We show that all half-BPS Wilson loop operators in N=4 SYM -- which are labeled by Young tableaus -- have a gravitational dual description in terms of D5-branes or alternatively in terms of D3-branes in AdS_5xS^5. We prove that the insertion of a half-BPS Wilson loop operator in the cal N=4 SYM path integral is achieved by integrating out the degrees of freedom on a configuration of bulk D5-branes or alternatively on a configuration of bulk D3-branes. The bulk D5-brane and D3...

In the holographic framework, a half BPS Wilson loop in ${\cal N}=4$ supersymmetric Yang-Mills in the fundamental, symmetric or antisymmetric representation of SU(N), is best described by a fundamental string, a D3 brane or a D5 brane with fluxes in their worldvolumes, respectively. We derive the spectrum of excitations of such D3 brane in $AdS_5\times S^5$ explicitly, considering its action in both the bosonic and the fermionic sectors, and demonstrate that it is organized a...

We present a theorem that determines the value of the Wilson loop associated with a Nambu-Goto action which generalizes the action of the $AdS_5\times S_5$ model. In particular we derive sufficient conditions for confining behavior. We then apply this theorem to various string models. We go beyond the classical string picture by incorporating quadratic quantum fluctuations. We show that the bosonic determinant of $D_p$ branes with 16 supersymmetries yields a Luscher term. W...

We systematically study the spectrum of excitations and the one-loop determinant of holographic Wilson loop operators in antisymmetric representations of $\mathcal{N}=4$ supersymmetric Yang-Mills theory. Holographically, these operators are described by D5-branes carrying electric flux and wrapping an $S^4 \subset S^5$ in the $AdS_5\times S^5$ bulk background. We derive the dynamics of both bosonic and fermionic excitations for such D5-branes. A particularly important configu...

These notes provide an introduction toward Wilson loops in N=4 supersymmetric Yang-Mills theory with a focus toward their integrability properties. In addition to a brief discussion of exact results for the circular Wilson loop and the cusp anomalous dimension, the notes focus on non-local symmetries, utilizing the integrability of the minimal surface problem that appears at strong coupling. This work is based on lectures given at the Young Researchers Integrability School an...